Sedimentation of generalized systems of interacting particles. III. Concentration-dependent sedimentation and extension to other transport methods

Hydrodynamic and thermodynamic analysis of PEGylated human serum albumin

Covalent labeling of therapeutic drugs and proteins with polyethylene glycol (PEGylation) is an important modification for improving stability, solubility, and half-life. PEGylation alters protein solution behavior through its impact on thermodynamic nonideality by increasing the excluded volume, and on hydrodynamic nonideality by increasing the frictional drag. To understand PEGylation's impact, we investigated the thermodynamic and hydrodynamic properties of a model system consisting of PEGylated human serum albumin derivatives using analytical ultracentrifugation (AUC) and dynamic light scattering (DLS). We constructed PEGylated human serum albumin derivatives of single, linear 5K, 10K, 20K, and 40K PEG chains and a single branched-chain PEG of 40K (2 × 20K). Sedimentation velocity (SV) experiments were analyzed using SEDANAL direct boundary fitting to extract ideal sedimentation coefficients so, hydrodynamic nonideality ks, and thermodynamic nonideality 2BM1SV terms. These quantities allow the determination of the Stokes radius Rs, the frictional ratio f/fo, and the swollen or entrained volume Vs/v, which measure size, shape, and solvent interaction. We performed sedimentation equilibrium experiments to obtain independent measurements of thermodynamic nonideality 2BM1SE. From DLS measurements, we determined the interaction parameter, kD, the concentration dependence of the apparent diffusion coefficient, D, and from extrapolation of D to c = 0 a second estimate of Rs. Rs values derived from SV and DLS measurements and ensemble model calculations (see complementary study) are then used to show that ks + kD = theoretical 2B22M1. In contrast, experimental BM1 values from SV and sedimentation equilibrium data collectively allow for similar analysis for protein-PEG conjugates and show that ks + kD = 1.02-1.07∗BM1, rather than the widely used ks + kD = 2BM1 developed for hard spheres. The random coil behavior of PEG dominates the colloidal properties of PEG-protein conjugates and exceeds the sum of a random coil and hard-sphere volume due to excess entrained water.

Energetic Dissection of Mab-Specific Reversible Self-Association Reveals Unique Thermodynamic Signatures

PURPOSE

Reversible self-association (RSA) remains a challenge in the development of therapeutic monoclonal antibodies (mAbs). We recently analyzed the energetics of RSA for five IgG mAbs (designated as A-E) under matched conditions and using orthogonal methods. Here we examine the thermodynamics of RSA for two of the mAbs that showed the strongest evidence of RSA (mAbs C and E) to identify underlying mechanisms.

METHODS

Concentration-dependent dynamic light scattering and sedimentation velocity (SV) studies were carried out for each mAb over a range of temperatures. Because self-association was weak, the SV data were globally analyzed via direct boundary fitting to identify best-fit models, accurately determine interaction energetics, and account for the confounding effects of thermodynamic and hydrodynamic nonideality.

RESULTS

mAb C undergoes isodesmic self-association at all temperatures examined, with the energetics indicative of an enthalpically-driven reaction offset by a significant entropic penalty. By contrast, mAb E undergoes monomer-dimer self-association, with the reaction being entropically-driven and comprised of only a small enthalpic contribution.

CONCLUSIONS

Classical interpretations implicate van der Waals interactions and H-bond formation for mAb C RSA, and electrostatic interactions for mAb E. However, noting that RSA is likely coupled to additional equilibria, we also discuss the limitations of such interpretations.

Analysis of nonideality: insights from high concentration simulations of sedimentation velocity data

The Aviv fluorescence detection system (Aviv-FDS) has allowed the performance of sedimentation velocity experiments on therapeutic antibodies in highly concentrated environments like formulation buffers and serum. Methods were implemented in the software package SEDANAL for the analysis of nonideal, weakly associating AUC data acquired on therapeutic antibodies and proteins (Wright et al. Eur Biophys J 47:709–722, 2018, Anal Biochem 550:72–83, 2018). This involved fitting both hydrodynamic, k_s, and thermodynamic, BM₁, nonideality where concentration dependence is expressed as s = s^o/(1 + k_sc) and D = D^o(1 + 2BM₁c)/(1 + k_sc) and s^o and D^o are values extrapolated to c = 0 (mg/ml). To gain insight into the consequences of these phenomenological parameters, we performed simulations with SEDANAL of a monoclonal antibody as a function of k_s (0–100 ml/g) and BM₁ (0–100 ml/g). This provides a visual understanding of the separate and joint impact of k_s and BM₁ on the shape of high-concentration sedimentation velocity boundaries and the challenge of their unique determination by finite element methods. In addition, mAbs undergo weak self- and hetero-association (Yang et al. Prot Sci 27:1334–1348, 2018) and thus we have simulated examples of nonideal weak association over a wide range of concentrations (1–120 mg/ml). Here we demonstrate these data are best analyzed by direct boundary global fitting to models that account for k_s, BM₁ and weak association. Because a typical clinical dose of mAb is 50–200 mg/ml, these results have relevance for biophysical understanding of concentrated therapeutic proteins.

Determination of Interaction Parameters for Reversibly Self-Associating Antibodies: A Comparative Analysis

Monoclonal antibodies (mAbs) represent a major class of biotherapeutics and are the fastest growing category of biologic drugs on the market. However, mAb development and formulation are often impeded by reversible self-association (RSA), defined as the dynamic exchange of monomers with native-state oligomers. Here, we present a comparative analysis of the self-association properties for 5 IgG mAbs, under matched conditions and using orthogonal methods. Concentration-dependent dynamic light scattering and sedimentation velocity studies revealed that the majority of mAbs examined exhibited weak to moderate RSA. However, because these studies were carried out at mAb concentrations in the mg/mL range, we also observed significant nonideality. Noting that nonideality frequently masks RSA and vice versa, we conducted direct boundary fitting of the sedimentation velocity data to determine stoichiometric binding models, interaction affinities, and nonideality terms for each mAb. These analyses revealed equilibrium constants from micromolar to millimolar and stoichiometric models from monomer-dimer to isodesmic. Moreover, even for those mAbs described by identical models, we observed distinct kinetics of self-association. The accuracy of the models and their corresponding equilibrium constants were addressed using sedimentation equilibrium and simulations. Overall, these results serve as the starting point for the comparative dissection of RSA mechanisms in therapeutic mAbs.

Sedimentation patterns of rapidly reversible protein interactions

The transport behavior of macromolecular mixtures with rapidly reversible complex formation is of great interest in the study of protein interactions by many different methods. Complicated transport patterns arise even for simple bimolecular reactions, when all species exhibit different migration velocities. Although partial differential equations are available to describe the spatial and temporal evolution of the interacting system given particular initial conditions, a general overview of the phase behavior of the systems in parameter space has not yet been reported. In the case of sedimentation of two-component mixtures, this study presents simple analytical solutions that solve the underlying equations in the diffusion-free limit previously subject to Gilbert-Jenkins theory. The new expressions describe, with high precision, the average sedimentation coefficients and composition of each boundary, which allow the examination of features of the whole parameter space at once, and may be used for experimental design and robust analysis of experimental boundary patterns to derive the stoichiometry and affinity of the complex. This study finds previously unrecognized features, including a phase transition between boundary patterns. The model reveals that the time-average velocities of all components in the reaction mixture must match-a condition that suggests an intuitive physical picture of an effective particle of the coupled cosedimentation of an interacting system. Adding to the existing numerical solutions of the relevant partial differential equations, the effective particle model provides physical insights into the relationships of the parameters that govern sedimentation patterns.

Protein-protein and ligand-protein interactions studied by analytical ultracentrifugation

All biological processes involve molecular interactions that result in either binding, self-association, or hetero-associations of one form or another. It is important to understand that no interactions are completely all-or-none. Some approach all-or-none only when there is strong positive cooperativity. Examples will be given of typical biomolecular interactions and their expected dependence on concentration, in order to point out the relatively wide range of concentration over which these types of phenomena take place. This chapter is concerned both with the binding of low-molecular-weight ligands to macromolecules as well as interactions between macromolecules using analytical ultracentrifugation (AUC) as a tool for measuring association properties of these systems. The theory of sedimentation of both ideal and nonideal interacting and noninteracting systems is discussed. Examples are given of each type of system along with a discussion of how each type of system can be analyzed. Several methods of data analysis are discussed.

Extracting equilibrium constants from kinetically limited reacting systems

It has been known for some time that slow kinetics will distort the shape of a reversible reaction boundary. Here we present a tutorial on direct boundary fitting of sedimentation velocity data for a monomer-dimer system that exhibits kinetic effects. Previous analysis of a monomer-dimer system suggested that rapid reaction behavior will persist until the relaxation time of the system exceeds 100 s (reviewed in Kegeles and Cann, 1978). Utilizing a kinetic integrator feature in Sedanal (Stafford and Sherwood, 2004), we can now fit for the k(off) values and measure the uncertainty at the 95% confidence interval. For the monomer-dimer system the range of well determined k(off) values is limited to 0.005 to 10(-5) s(-1) corresponding to relaxation times (at a loading concentration of the Kd) of approximately 70 to approximately 33,000 s. For shorter relaxation times the system is fast and only the equilibrium constant K but not k(off) can be uniquely determined. For longer relaxation times the system is irreversibly slow, and assuming the system was at initial equilibrium before the start of the run, only the equilibrium constant K but not k(off) can be uniquely determined.

A new adaptive grid-size algorithm for the simulation of sedimentation velocity profiles in analytical ultracentrifugation

Analytical ultracentrifugation allows one to measure in real-time the concentration gradients arising from the application of a centrifugal force to macromolecular mixtures in solution. In the last decade, the ability to efficiently solve the partial differential equation governing the ultracentrifugal sedimentation and diffusion process, the Lamm equation, has spawned significant progress in the application of sedimentation velocity analytical ultracentrifugation for the study of biological macromolecules, for example, the characterization of protein oligomeric states and the study of reversible multi-protein complexes in solution. The present work describes a numerical algorithm that can provide an improvement in accuracy or efficiency over existing algorithms by more than one order of magnitude, and thereby greatly facilitate the practical application of sedimentation velocity analysis, in particular, for the study of multi-component macromolecular mixtures. It is implemented in the public domain software SEDFIT for the analysis of experimental data.

Analysis of heterologous interacting systems by sedimentation velocity: curve fitting algorithms for estimation of sedimentation coefficients, equilibrium and kinetic constants

Analytical ultracentrifugation (AUC) has played and will continue to play an important role in the investigation of protein-protein, protein-DNA and protein-ligand interactions. A major advantage of AUC over other methods is that it allows the analysis of systems free in solution in nearly any buffer without worry about spurious interactions with a supporting matrix. Large amounts of high-quality data can be acquired in relatively short times. Advances in software for the treatment of AUC data over the last decade have eliminated many of the tedious aspects of AUC data analysis, allowing relatively rapid analysis of complicated systems that were previously unapproachable. A software package called sedanal is described that can perform global fits to AUC sedimentation velocity data obtained for both interacting and non-interacting, macromolecular multi-species, multi-component systems, by combining data from multiple runs over a range of sample concentrations and component ratios. Interaction parameters include both forward and reverse rate constants, or equilibrium constants, for each reaction, as well as concentration dependence of both sedimentation and diffusion coefficients. sedanal fits to time-difference data to eliminate time-independent systematic errors inherent in AUC data. The sedanal software package is based on the use of finite-element numerical solutions of the Lamm equation.

A model for sedimentation in inhomogeneous media. I. Dynamic density gradients from sedimenting co-solutes

Macromolecular sedimentation in inhomogeneous media is of great practical importance. Dynamic density gradients have a long tradition in analytical ultracentrifugation, and are frequently used in preparative ultracentrifugation. In this paper, a new theoretical model for sedimentation in inhomogeneous media is presented, based on finite element solutions of the Lamm equation with spatial and temporal variation of the local solvent density and viscosity. It is applied to macromolecular sedimentation in the presence of a dynamic density gradient formed by the sedimentation of a co-solute at high concentration. It is implemented in the software SEDFIT for the analysis of experimental macromolecular concentration distributions. The model agrees well with the measured sedimentation profiles of a protein in a dynamic cesium chloride gradient, and may provide a measure for the effects of hydration or preferential solvation parameters. General features of protein sedimentation in dynamic density gradients are described.

A Comparison of the Self-association Behavior of the Plant Cyclotides Kalata B1 and Kalata B2 via Analytical Ultracentrifugation

The recently discovered cyclotides kalata B1 and kalata B2 are miniproteins containing a head-to-tail cyclized backbone and a cystine knot motif, in which disulfide bonds and the connecting backbone segments form a ring that is penetrated by the third disulfide bond. This arrangement renders the cyclotides extremely stable against thermal and enzymatic decay, making them a possible template onto which functionalities can be grafted. We have compared the hydrodynamic properties of two prototypic cyclotides, kalata B1 and kalata B2, using analytical ultracentrifugation techniques. Direct evidence for oligomerization of kalata B2 was shown by sedimentation velocity experiments in which a method for determining size distribution of polydisperse molecules in solution was employed. The shape of the oligomers appears to be spherical. Both sedimentation velocity and equilibrium experiments indicate that in phosphate buffer kalata B1 exists mainly as a monomer, even at millimolar concentrations. In contrast, at 1.6 mm, kalata B2 exists as an equilibrium mixture of monomer (30%), tetramer (42%), octamer (25%), and possibly a small proportion of higher oligomers. The results from the sedimentation equilibrium experiments show that this self-association is concentration dependent and reversible. We link our findings to the three-dimensional structures of both cyclotides, and propose two putative interaction interfaces on opposite sides of the kalata B2 molecule, one involving a hydrophobic interaction with the Phe6, and the second involving a charge-charge interaction with the Asp25 residue. An understanding of the factors affecting solution aggregation is of vital importance for future pharmaceutical application of these molecules.

On the analysis of protein self-association by sedimentation velocity analytical ultracentrifugation

Analytical ultracentrifugation is one of the classical techniques for the study of protein interactions and protein self-association. Recent instrumental and computational developments have significantly enhanced this methodology. In this paper, new tools for the analysis of protein self-association by sedimentation velocity are developed, their statistical properties are examined, and considerations for optimal experimental design are discussed. A traditional strategy is the analysis of the isotherm of weight-average sedimentation coefficients s(w) as a function of protein concentration. From theoretical considerations, it is shown that integration of any differential sedimentation coefficient distribution c(s), ls-g(*)(s), or g(s(*)) can give a thermodynamically well-defined isotherm, as long as it provides a good model for the sedimentation profiles. To test this condition for the g(s(*)) distribution, a back-transform into the original data space is proposed. Deconvoluting diffusion in the sedimentation coefficient distribution c(s) can be advantageous to identify species that do not participate in the association. Because of the large number of scans that can be analyzed in the c(s) approach, its s(w) values are very precise and allow extension of the isotherm to very low concentrations. For all differential sedimentation coefficients, corrections are derived for the slowing of the sedimentation boundaries caused by radial dilution. As an alternative to the interpretation of the isotherm of the weight-average s value, direct global modeling of several sedimentation experiments with Lamm equation solutions was studied. For this purpose, a new software SEDPHAT is introduced, allowing the global analysis of several sedimentation velocity and equilibrium experiments. In this approach, information from the shape of the sedimentation profiles is exploited, which permits the identification of the association scheme and requires fewer experiments to precisely characterize the association. Further, under suitable conditions, fractions of incompetent material that are not part of the reversible equilibrium can be detected.

Non-ideality by sedimentation velocity of halophilic malate dehydrogenase in complex solvents

We have investigated the potential of sedimentation velocity analytical ultracentrifugation for the measurement of the second virial coefficients of proteins, with the goal of developing a method that allows efficient screening of different solvent conditions. This may be useful for the study of protein crystallization. Macromolecular concentration distributions were modeled using the Lamm equation with the approximation of linear concentration dependencies of the diffusion constant, D = D(o) (1 + k(D)c), and the reciprocal sedimentation coefficient s = s(o)/(1 + k(s)c). We have studied model distributions for their information content with respect to the particle and its non-ideal behavior, developed a strategy for their analysis by direct boundary modeling, and applied it to data from sedimentation velocity experiments on halophilic malate dehydrogenase in complex aqueous solvents containing sodium chloride and 2-methyl-2,4-pentanediol, including conditions near phase separation. Using global modeling for three sets of data obtained at three different protein concentrations, very good estimates for k(s) and s degrees and also for D degrees and the buoyant molar mass were obtained. It was also possible to obtain good estimates for k(D) and the second virial coefficients. Modeling of sedimentation velocity profiles with the non-ideal Lamm equation appears as a good technique to investigate weak inter-particle interactions in complex solvents and also to extrapolate the ideal behavior of the particle.

Self-association of human apolipoprotein E3 and E4 in the presence and absence of phospholipid

Human apolipoprotein E (apoE) exists as three main isoforms, differing by single amino acid substitutions, with the apoE4 isoform strongly linked to the incidence of late onset Alzheimer's disease. We have expressed and purified apoE3 and apoE4 from Escherichia coli and compared their hydrodynamic properties by gel permeation liquid chromatography, capillary electrophoresis, circular dichroism, and sedimentation methods. Sedimentation velocity experiments, employing a new method for determining the size distribution of polydisperse macromolecules in solution (Schuck, P. (2000) Biophys. J. 78, 1606-1619), provide direct evidence for the heterogeneous solution structures of apoE3 and apoE4. In a lipid-free environment, apoE3 and apoE4 exist as a slow equilibrium mixture of monomer, tetramer, octamer, and a small proportion of higher oligomers. Both sedimentation velocity and equilibrium experiments indicate that apoE4 has a greater propensity to self-associate. We also demonstrate that apoE3 and apoE4 oligomers dissociate significantly in the presence of dihexanoylphosphatidylcholine micelles (20 mm) and to a lesser extent at submicellar concentrations (4 mm). The alpha-helical content for both isoforms was almost identical (50%) in the presence and absence of dihexanoylphosphatidylcholine. These results reveal that apoE oligomers undergo phospholipid-induced dissociation to folded monomers, suggesting the monomeric form prevails on the lipoprotein surface in vivo.

Size-distribution analysis of macromolecules by sedimentation velocity ultracentrifugation and lamm equation modeling

A new method for the size-distribution analysis of polymers by sedimentation velocity analytical ultracentrifugation is described. It exploits the ability of Lamm equation modeling to discriminate between the spreading of the sedimentation boundary arising from sample heterogeneity and from diffusion. Finite element solutions of the Lamm equation for a large number of discrete noninteracting species are combined with maximum entropy regularization to represent a continuous size-distribution. As in the program CONTIN, the parameter governing the regularization constraint is adjusted by variance analysis to a predefined confidence level. Estimates of the partial specific volume and the frictional ratio of the macromolecules are used to calculate the diffusion coefficients, resulting in relatively high-resolution sedimentation coefficient distributions c(s) or molar mass distributions c(M). It can be applied to interference optical data that exhibit systematic noise components, and it does not require solution or solvent plateaus to be established. More details on the size-distribution can be obtained than from van Holde-Weischet analysis. The sensitivity to the values of the regularization parameter and to the shape parameters is explored with the help of simulated sedimentation data of discrete and continuous model size distributions, and by applications to experimental data of continuous and discrete protein mixtures.

Direct sedimentation analysis of interference optical data in analytical ultracentrifugation

Sedimentation data acquired with the interference optical scanning system of the Optima XL-I analytical ultracentrifuge can exhibit time-invariant noise components, as well as small radial-invariant baseline offsets, both superimposed onto the radial fringe shift data resulting from the macromolecular solute distribution. A well-established method for the interpretation of such ultracentrifugation data is based on the analysis of time-differences of the measured fringe profiles, such as employed in the g(s*) method. We demonstrate how the technique of separation of linear and nonlinear parameters can be used in the modeling of interference data by unraveling the time-invariant and radial-invariant noise components. This allows the direct application of the recently developed approximate analytical and numerical solutions of the Lamm equation to the analysis of interference optical fringe profiles. The presented method is statistically advantageous since it does not require the differentiation of the data and the model functions. The method is demonstrated on experimental data and compared with the results of a g(s*) analysis. It is also demonstrated that the calculation of time-invariant noise components can be useful in the analysis of absorbance optical data. They can be extracted from data acquired during the approach to equilibrium, and can be used to increase the reliability of the results obtained from a sedimentation equilibrium analysis.

Sedimentation analysis of noninteracting and self-associating solutes using numerical solutions to the Lamm equation

The potential of using the Lamm equation in the analysis of hydrodynamic shape and gross conformation of proteins and reversibly formed protein complexes from analytical ultracentrifugation data was investigated. An efficient numerical solution of the Lamm equation for noninteracting and rapidly self-associating proteins by using combined finite-element and moving grid techniques is described. It has been implemented for noninteracting solutes and monomer-dimer and monomer-trimer equilibria. To predict its utility, the error surface of a nonlinear regression of simulated sedimentation profiles was explored. Error contour maps were calculated for conventional independent and global analyses of experiments with noninteracting solutes and with monomer-dimer systems at different solution column heights, loading concentrations, and centrifugal fields. It was found that the rotor speed is the major determinant for the shape of the error surface, and that global analysis of different experiments can allow substantially improved characterization of the solutes. We suggest that the global analysis of the approach to equilibrium in a short-column sedimentation equilibrium experiment followed by a high-speed short-column sedimentation velocity experiment can result in sedimentation and diffusion coefficients of very high statistical accuracy. In addition, in the case of a protein in rapid monomer-dimer equilibrium, this configuration was found to reveal the most precise estimate of the association constant.

Determination of molecular parameters by fitting sedimentation data to finite-element solutions of the Lamm equation

A method for fitting experimental sedimentation velocity data to finite-element solutions of various models based on the Lamm equation is presented. The method provides initial parameter estimates and guides the user in choosing an appropriate model for the analysis by preprocessing the data with the G(s) method by van Holde and Weischet. For a mixture of multiple solutes in a sample, the method returns the concentrations, the sedimentation (s) and diffusion coefficients (D), and thus the molecular weights (MW) for all solutes, provided the partial specific volumes (v) are known. For nonideal samples displaying concentration-dependent solution behavior, concentration dependency parameters for s(sigma) and D(delta) can be determined. The finite-element solution of the Lamm equation used for this study provides a numerical solution to the differential equation, and does not require empirically adjusted correction terms or any assumptions such as infinitely long cells. Consequently, experimental data from samples that neither clear the meniscus nor exhibit clearly defined plateau absorbances, as well as data from approach-to-equilibrium experiments, can be analyzed with this method with enhanced accuracy when compared to other available methods. The nonlinear least-squares fitting process was accomplished by the use of an adapted version of the "Doesn't Use Derivatives" nonlinear least-squares fitting routine. The effectiveness of the approach is illustrated with experimental data obtained from protein and DNA samples. Where applicable, results are compared to methods utilizing analytical solutions of approximated Lamm equations.

Identification and interpretation of complexity in sedimentation velocity boundaries

Synthetic sedimentation velocity boundaries were generated using finite-element solutions to the original and modified forms of the Lamm equation. Situations modeled included ideal single- and multicomponent samples, concentration-dependent samples, noninteracting multicomponent samples, and reversibly self-associating samples. Synthetic boundaries subsequently were analyzed using the method of van Holde and Weischet, and results were compared against known input parameters. Results indicate that this analytical method provides rigorous diagnostics for virtually every type of sample complexity encountered experimentally. Accordingly, both the power and utility of sedimentation velocity experiments have been significantly expanded.

Kinetic analysis of the hydrodynamic transition accompanying protein folding using size exclusion chromatography. 1. Denaturant dependent baseline changes

Size exclusion profiles of proteins with persistent conformations exhibit broad asymmetric peaks whose shape and elution times are dependent on denaturant concentration. The collective elution profiles were precisely simulated by an apparent binding model that treats the denaturant dependence in terms of an apparent matrix binding. The model requires three experimentally measurable parameters: the elution time for the unbound protein, an apparent association equilibrium constant for binding, and an apparent exchange time for binding. The denaturant dependence for each of these parameters is related to the accessible surface area of the protein.

Generalized finite element solution to one-dimensional flux problems

A finite element numerical solution to the general one-dimensional flow equation is derived in a form that provides a convenient and general means to simulate a wide variety of one-dimensional flow techniques of interest to biological scientists, e.g., ultracentrifugation, electrophoresis, chromatography, etc. Diverse physical models defined in terms of column geometry, solute interactions, and the dependence of transport parameters on column position, time, or concentrations of one or more solutes, can be accommodated. A particularly useful aspect of the formulation is that a wide variety of boundary conditions can be simply applied to the end result, without rederivation of the solution for each new case. The numerical solution is expressed as matrix equations that are sufficiently general so that incorporation of particular models can be effected by substitution of appropriate quantities into the final result.

General solution to the inverse problem of the differential equation of the ultracentrifuge

Whenever experimental data can be simulated according to a model of the physical process, values of physical parameters in the model can be determined from experimental data by use of a nonlinear least-squares algorithm. We have used this principle to obtain a general procedure for evaluating molecular parameters of solutes redistributing in the ultracentrifuge that uses time-dependent concentration, concentration-difference, or concentration-gradient data. The method gives the parameter values that minimize the sum of the squared differences between experimental data and simulated data calculated from numerical solutions to the differential equation of the ultracentrifuge.

Catalytically active monomer and dimer forms of rat liver carbamoyl-phosphate synthetase

Purified carbamoyl-phosphate synthetase of rat liver is shown to exist in a state of rapid, reversible monomer-dimer equilibrium. The allosteric activator N-acetyl-L-glutamate displaces the equilibrium toward monomer formation. This effect is observed over a range of initial protein concentrations of 0.02-5 mg/mL. Measurements of Stokes radii by analytical gel chromatography indicate that at concentrations less than 0.1 mg/mL at 25 degrees C in the presence of all the substrates the enzyme exists as a monomer of 160000 molecular weight. A gel chromatographic method was developed to identify the active form of carbamoyl-phosphate synthetase. On the basis of analysis of the ADP boundary formed during gel chromatography, the monomer is established to be catalytically active. Active enzyme centrifugation studies confirm that the monomer is a reactive species and suggest that the dimer also functions catalytically. Under the conditions of the usual enzyme assay, carbamoyl-phosphate synthetase is mainly in the monomer form. Activation by acetylglutamate can occur at the level of the monomer and is not coupled to dissociation since the enzyme dissociates at low concentrations even in the absence of acetylglutamate. The stoichiometry of the association is observed directly in the electron microscope. The dimensions of the negatively stained particles of the enzyme in the presence or absence of substrates correspond to monomers and dimers, assuming the molecule to be a prolate ellipse. The number of monomers observed in the presence of substrate represents 86% of the total number of enzyme molecules. The average molecular weight calculated from the numbers of particles seen in negatively stained specimens of carbamoyl-phosphate synthetase is 182000. Electron microscope studies provide independent evidence for monomer--dimer interactions and show that under the conditions examined the enzyme is mainly in the monomer form.